First of all the definition of irreducibility:
- A topological space $X$ is called reducible if $X$ can be written as a union of two non-empty and closed proper subsets of $X$. We call $X$ irreducible if it is not reducible.
- A subset $F$ of $X$ is called reducible resp. irreducible if it has this property in its subspace topology.
Now on Wikipedia I have found the following definition of irreducible component:
- An irreducible component of a topological space is a maximal irreducible subset.
My question: What do we mean with maximal here? I thought if a subset is irreducible it is automatically maximal because we can not split it up into two smaller closed sets. Can someone explain what is wrong with my thought? It would be great if someone could give me an specific example (e.g. from the Zariski topology).
I also have no idea why irreducible components are also closed but I hope that I will understand it if I could solve my problem above.
Thank you in advance.